Arkiv för Matematik

, Volume 35, Issue 2, pp 201-224

First online:

Wiggly sets and limit sets

  • Christopher J. BishopAffiliated withMathematics Department, State University of New York at Stony Brook
  • , Peter W. JonesAffiliated withMathematics Department, Yale University

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We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that ifG is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1.